Separable elastic Banach spaces are universal

2015/02/12 by Dale E. Alspach, Alspach, Dale E., Bunyamin Sari +1
Mathematics · #46B03 (primary) #46B25 (secondary) #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:46B03 #msc:46B25

paper · pdf · doi:10.48550/arxiv.1502.03791

27 pages

arxiv created 2015/02/12 · arxiv updated 2015/02/13

Abstract

A Banach space X is elastic if there is a constant K so that whenever a Banach space Y embeds into X, then there is an embedding of Y into X with constant K. We prove that C[0,1] embeds into separable infinite dimensional elastic Banach spaces, and therefore they are universal for all separable Banach spaces. This confirms a conjecture of Johnson and Odell. The proof uses incremental embeddings into X of C(K) spaces for countable compact K of increasing complexity. To achieve this we develop a generalization of Bourgain's basis index that applies to unconditional sums of Banach spaces and prove a strengthening of the weak injectivity property of these C(K) that is realized on special reproducible bases.

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